There exist less methods for the harder problem of under-approximating images of functions or sets of reachable states. Some approaches have been proposed for linear discrete-time systems [1]}, [2]}, [3]}.
Interval-based methods, relying on space discretization, have been used for under-approximating the image of nonlinear functions [4]}. They were also used to over and under approximate solutions of differential systems with uncertain initial conditions [5]}.
Tight approximations for reachable sets of nonlinear continuous systems can be found via expensive Eulerian methods: the zero sub-level set of the Lipschitz viscosity solution to a Hamilton-Jacobi (HJB) partial differential equation gives the (backward) reachable set [6]}. Other approaches, using SoS methods and LMI relaxations have been proposed for inner approximations, see e.g. [7]}. In [8]}, under-approximations for polynomial systems are obtained by solving semi-definite programs.
Taylor models are used on the inverse flow map to derive under-approximations [9]}, but using topological conditions that are checked with interval constraints solving, which have difficulties to scale up with dimension.
In [10]}, the computation of the under-approximated reachable set is based on a costly analysis of the boundary of the reachable sets and polytopic approximations.
In [11]}, some non-convex under-approximations are computed with polynomial zonotopes, relying on a computation of the outer-approximation of the reachable set, of an enclosure of the boundary of the reachable set, and a reduction of the outer-approximation until it is fully included in the region delimited by the boundary.
